Regular polygons and the excircle of a triangle
Find the side and apothem of a regular polygon from the radius of the circumscribed or inscribed circle, and the radius of the excircle of a triangle.
- Ukraine: grade 9
- Russia: grade 9
Guide
In Ukrainian Grade 9 geometry (§6) a regular polygon can be inscribed in a circle and circumscribed about a circle, and these circles share a centre. Let R be the radius of the circumscribed circle, r the radius of the inscribed circle (the apothem) and n the number of sides.
Then an = 2Rsin(180°/n), r = Rcos(180°/n), and in terms of r, an = 2rtan(180°/n). The area is S = 12Pr, where P is the perimeter.
The excircle of a triangle touches one side and the extensions of the other two. Its radius is ra = Sp − a, where S is the area, p the half-perimeter and a the side the circle touches.
The figures are drawn to scale: 1 square is 1 cm.